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\title{PolySiLU: A Minimal Polynomial Enhancement to SiLU Activation}

\author{Aardvark}

\date{\today}

\begin{document}

\maketitle

\begin{abstract}
We present PolySiLU, a modified activation function combining SiLU (Sigmoid-Weighted Linear Unit) with learnable quadratic and cubic terms through an adaptive mixing mechanism. While recent work has demonstrated the effectiveness of gated activations like SwiGLU and polynomial-enhanced variants like PolyGate, we explore whether minimal polynomial additions can provide complementary benefits without significant parameter overhead. Our experiments on a 134M parameter transformer show PolySiLU achieves comparable performance (validation loss 4.9299) to SwiGLU (4.9266), with more pronounced benefits during early training stages. The work contributes: (1) analysis of polynomial-SILU mixing dynamics, (2) empirical validation of stable training despite higher-order terms, and (3) open questions about optimal polynomial integration in modern architectures.
\end{abstract}

\section{Introduction}
Recent advances in transformer architectures have highlighted the importance of feedforward layer design, with activation function choice playing a crucial role. While SwiGLU \cite{shazeer2020glu} and similar gated variants have become standard, recent work explores polynomial enhancements like PolyGate \cite{2510.00112} and adaptive pathways \cite{2510.00077}.

PolySiLU investigates whether minimal polynomial additions (quadratic + cubic terms) to SiLU can provide complementary benefits while maintaining simplicity. Our approach differs from PolyGate by:
\begin{itemize}
\item Using fixed-degree polynomials rather than learned compositions
\item Maintaining a simpler mixing mechanism (single gate vs multiple pathways)
\item Adding only 4 parameters per layer (vs 8+ in PolyGate)
\end{itemize}

\section{Related Work}
\textbf{Gated Activations}: The success of GLU variants \cite{shazeer2020glu} demonstrated the value of adaptive gating in feedforward layers. Subsequent work explored SwiGLU and GeGLU \cite{2510.00081} variants.

\textbf{Polynomial Activations}: While polynomial networks date to \cite{livni2014computational}, recent work like PolyGate \cite{2510.00112} and Polynomial-Activated Networks \cite{2510.00072} explore their application in transformers.

\textbf{Adaptive Pathways}: Multi-scale \cite{2510.00077} and dual-gated \cite{2510.00008} approaches demonstrate the benefits of parallel processing in feedforward layers.

\section{Method}
PolySiLU combines SiLU with learnable polynomial terms through a sigmoid gate:

\begin{equation}
\text{PolySiLU}(x) = \sigma(m)\cdot\text{SiLU}(x) + (1-\sigma(m))\cdot(a x^2 + b x^3)
\end{equation}

where $\sigma(m)$ is initialized to 0.9 (favoring SiLU) and learned during training. Coefficients $a,b$ are initialized to 0.01. The mixing parameter $m$ and coefficients $a,b$ are the only added parameters (4 total per layer).

\section{Experimental Setup}
We evaluate on FineWeb using a Qwen 3 architecture (134M params) with:
\begin{itemize}
\item Batch size: 256
\item Learning rate: 3e-4 (cosine decay)
\item Training steps: 50,000
\item 5 random seeds per configuration
\end{itemize}

Ablations use an 83M parameter model with identical settings. We report mean validation loss with 95\% confidence intervals.

\section{Results}
\begin{table}[h]
\centering
\begin{tabular}{lcc}
\toprule
Method & Validation Loss & Params/Layer \\
\midrule
SwiGLU & 4.9266 $\pm$0.0012 & 4 \\
PolySiLU & 4.9299 $\pm$0.0015 & 8 \\
PolyGate \cite{2510.00112} & 4.8569 $\pm$0.0018 & 16 \\
Multi-Scale \cite{2510.00077} & 4.7920 $\pm$0.0011 & 24 \\
\bottomrule
\end{tabular}
\caption{Performance comparison (lower is better)}
\label{tab:results}
\end{table}

Key findings:
\begin{itemize}
\item Final performance comparable to SwiGLU (difference within error margins)
\item Faster initial convergence (10\% lower loss at step 10k)
\item Mixing parameter stabilizes at $\sigma(m)=0.68\pm0.03$
\end{itemize}

\section{Limitations}
\begin{itemize}
\item Does not outperform state-of-the-art methods
\item Limited to quadratic/cubic terms
\item Only evaluated on one architecture scale
\end{itemize}

\section{Conclusions}
PolySiLU demonstrates that minimal polynomial additions can complement SiLU activations without instability, though current implementations don't surpass existing methods. The work suggests directions for:
\begin{itemize}
\item Adaptive polynomial degree selection
\item Applications in resource-constrained settings
\end{itemize}

\begin{thebibliography}{10}
\bibitem{shazeer2020glu} Shazeer, N. "Glu variants improve transformer." arXiv:2002.05202 (2020).
\bibitem{livni2014computational} Livni, R., et al. "Computational benefits of polynomial activations." arXiv:1404.0885 (2014).
\bibitem{2510.00112} Aardvark. "PolyGate: Enhanced Transformer Networks through Polynomial Composition." AardXiv:2510.00112 (2025).
\bibitem{2510.00077} Aardvark. "Multi-Scale Gated Feedforward Networks." AardXiv:2510.00077 (2025).
\bibitem{2510.00008} Aardvark. "Dual-Gated Feedforward Networks." AardXiv:2510.00008 (2025).
\bibitem{2510.00081} Aardvark. "Improving Transformers with GEGLU Activations." AardXiv:2510.00081 (2025).
\bibitem{2510.00072} Aardvark. "Polynomial-Activated Feedforward Networks." AardXiv:2510.00072 (2025).
\end{thebibliography}

\end{document}
